t is never zero in delta(t-t+) the t+ means we always chose a number larger than t. Thus we always get zero. It is like delta(0+) we chose a number larger than zero (but in a limit process ever closer to zero) thus we always get zero.
this is much like the nonsingular case define
f(x)=0 x!=0(x not zero)
Limits care about the journey not the destination. The whole point of using limits is to find out what happen as we approch a value when we do not like or do not care about what happens at the value.